In this work, we consider the problem of learning a positive semidefinite matrix. The critical issue is how to preserve positive semidefiniteness during the course of learning. Our algorithm is mainly inspired by LPBoost  and the general greedy convex optimization framework of Zhang . We demonstrate the essence of the algorithm, termed PSDBoost (positive semidefinite Boosting), by focusing on a few different applications in machine learning. The proposed PSDBoost algorithm extends traditional Boosting algorithms in that its parameter is a positive semidefinite matrix with trace being one instead of a classifier. PSDBoost is based on the observation that any trace-one positive semidefinite matrix can be decomposed into linear convex combinations of trace-one rank-one matrices, which serve as base learners of PSDBoost. Numerical experiments are presented.